Wijsman and Hit-and-Miss Topologies of Quasi-Metric Spaces
- Cite this article as:
- Rodríguez-López, J. & Romaguera, S. Set-Valued Analysis (2003) 11: 323. doi:10.1023/A:1025675400451
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The relationship between the Wijsman topology and (proximal) hit-and-miss topologies is studied in the realm of quasi-metric spaces. We establish the equivalence between these hypertopologies in terms of Urysohn families of sets. Our results generalize well-known theorems and provide easier proofs. In particular, we prove that for a quasi-pseudo-metrizable space (X,T) the Vietoris topology on the set P0(X) of all nonempty subsets of X is the supremum of all Wijsman topologies associated with quasi-pseudo-metrics compatible with T. We also show that for a quasi-pseudo-metric space (X,d) the Hausdorff extended quasi-pseudo-metric is compatible with the Wijsman topology on P0(X) if and only if d−1 is hereditarily precompact.